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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a natural number (also called counting number) can mean either an element of the set = *n = = ? = (n-1) ?

and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n = m (in the naïve sense) if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 n is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of m.


Also, with this definition, different possible interpretations of notations like Rn (n-tuples versus mappings of n into R) coincide.


Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a natural number (also called counting number) can mean either an element of the set = *n = = ? = (n-1) ?

and so on. When a natural number is used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n = m (in the naïve sense) if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 n is a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of m.


Also, with this definition, different possible interpretations of notations like Rn (n-tuples versus mappings of n into R) coincide.


Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set n is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements.


Other constructions
Although the standard construction is useful, it is not the only possible construction. For example:
one could define 0 =
and S(a) = ,
producing
0 = 1 = = 2 = = , etc. Or we could even define 0 =
and S(a) = a U
producing
0 = 1 = 2 = Arguably the oldest set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell
Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, Order of Merit , Fellow of the Royal Society , was a British people philosopher, mathematical logic, mathematician, historian, advocate for social reform, and pacifism....
 under which each concrete natural number n is defined as the set of all sets with n elements. This may appear circular, but can be made rigorous with care. Define 0 as (clearly the set of all sets with 0 elements) and define (for any set A) as (see set-builder notation
Set-builder notation

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a Set by stating the properties that its members must satisfy....
). Then 0 will be the set of all sets with 0 elements, will be the set of all sets with 1 element, will be the set of all sets with 2 elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under (that is, if the set contains an element n, it also contains ). This definition does not work in the usual systems of axiomatic set theory because the collections involved are too large (it will not work in any set theory with the axiom of separation); but it does work in New Foundations
New Foundations

In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica....
 (and in related systems known to be consistent) and in some systems of type theory
Type theory

In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general....
.

See also

  • Integer
    Integer

    The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
  • Whole number
    Whole number

    The term whole number is used by various authors to mean either:*the nonnegative integer *the positive integer *all integer ...
  • Negative and non-negative numbers
    Negative and non-negative numbers

    A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
  • Countable set
    Countable set

    In mathematics, a countable set is a Set with the same cardinality as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....

External links

  • by Richard Dedekind
    Richard Dedekind

    Julius Wilhelm Richard Dedekind was a Germany mathematics who did important work in abstract algebra, algebraic number theory and the foundations of the real numbers....
     at Project Gutenberg
    Project Gutenberg

    Project Gutenberg, abbreviated as PG, is a volunteer effort to digitize, archive and distribute cultural works, as founder Michael Hart said "To encourage the creation and distribution of eBooks."....